Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in N dimensions

Abstract

This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+u(a−μur−1−λw),x∈Ω,t>0,τvt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0. Here, Ω⊂RN(N≥1) is a bounded domain with smooth boundary and τ>0,r>1,λ≥0, a∈R, μ,ξ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases r>2 or r=2, with μ>μ⁎=(N−2)+N(χ+Cβ)CN2+11N2+1 for some positive constants Cβ and CN2+1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((aμ)1r−1,(aμ)1r−1,0) if μ is large enough. This result improves or extends previous results of several authors.